In a neutron lifetime measurement at the Japan Proton Accelerator Complex, the neutron lifetime is calculated from the neutron decay rate and the incident neutron flux. The flux is obtained by counting the protons emitted from the neutron absorption reaction of 3He gas, which is diluted in a mixture of working gas in a detector. Hence, it is crucial to determine the amount of 3He in the mixture. In order to improve the accuracy of the number density of the 3He nuclei, we have suggested using the 14N(n, p)14C reaction as a reference because this reaction involves similar kinetic energy to the 3He(n,p)3 H reaction and a smaller reaction cross section to introduce reasonable large partial pressure. The uncertainty of the recommended value of the cross section, however, is not satisfied with our requirement. In this paper we report the most accurate experimental value of the cross section of the 14N(n,p)14C reaction at a neutron velocity of 2200 m s-1, measured relative to the 3He(n,p)3H reaction. The result was 1.868 \pm 0.003 (stat.) ± 0.006 (sys.) b. Additionally, the cross section of the 17O(n,α)14C reaction at the neutron velocity is also redetermined as 249 ± 6 mb.